A stochastic, or random, process describes the correlation or evolution of random events. It is used to model stock market fluctuations and electronic/audio-visual/biological signals. Among the most well-known stochastic processes are random walks and Brownian motion.

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Deriving master equation for discrete process

Consider a group of $N$ professors, $Y$ of whom are wearing white socks and $X = N − Y$ others who are wearing black socks. On each time step, one professor is chosen at random and he has to put a new ...
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15 views

what does it mean that a system is attractive?

What does it mean that a system is attractive in the context of Statistical Mechanics? Is this notion related to the presence of some monotonicity properties?
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15 views

Two Poisson processes

Suppose we have two independent homogeneous Poisson processes with intensities $a$ and $b$. What is the distribution the number of arrivals of the first process strictly between $k$-th and $m$-th ...
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14 views

Homogeneous Poisson process conditional expectation

I have a homogeneous Poisson Process with intensity $\lambda$, $S_n$ time of $n$-th arrival, $N_t$ number of arrivals till time $t$. How to compute $E(S_1\mid N_1\ge 2)$?
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1answer
26 views

Is Brownian Motion increasing?

Given a process $Y_t = e^{B_t}$ We know that since Brownian motion is continuous for $t \geq 0$. Since $B_t$ is a completely random motion, it is true that we cannot say whether it is monotone ...
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1answer
21 views

How to calculate $\mathbb{E}((B_3-B_2)(B_4-B_{\pi}) \mid B_1)$ for a Brownian motion $(B_t)_{t \geq 0}$

I want to find the best predictor of $(B_3-B_2)(B_4-B_{\pi})$ given an observation of $B_1$ Where $B_t$ is brownian motion for time $t \geq 0$. I am not sure how to approach this. I know it will be ...
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Best predictor of Brownian motion

Let $B_t$ be brownian motion at time $ t \geq 0$. Then I want to find the best predictor of $B_8 + 4$ given that there are observations of brownian motion up to time $t = 1$. Approach: Essentially, ...
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1answer
11 views

Conditional expectation and brownian motion - check my answer please

$X = \frac{ B_1+ B_3 - B_2}{\sqrt{2}}$ and $Y = \frac{B_1 - B_3+ B_2}{\sqrt{2}}$ Where $B_t$ Is brownian motion at time $t\geq0$ I want to find $\mathbb{E} [Y + 3X | X]$ It is known to me that $X, ...
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Using Feynamn-Kac to solve expectation

I want to solve an expectation $u(t,x)=\mathbb{E}[exp(\beta X_T)|\mathcal{F}_t]$ related to a random variable $X$ which statisfied the CIR process $$dX_s=k(\alpha-X_s)ds+\sigma\sqrt{X_s}dW_s$$, which ...
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1answer
15 views

Independence of two random variables derived from a Brownian motion

If $X = B_1 + B_3 - B_2$ and $Y = B_1 - B_3 + B_2$ Where $B_t$ is Brownian Motion for $t \geq 0$ And I want to state with certainty whether $X$ and $Y$ are indep or not, do I simply just ...
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1answer
31 views

Distribution of Brownian Motion help

If $X = \frac{B_1 - B_3 + B_2}{\sqrt{2}}$ Where $B_t$ is brownian motion at time $t$. And I want to find the the distribution of $X$, how would I do so? $E[X] = 0$ is fairly straight forward. For ...
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21 views

Girsanov's theorem and absolutely continuous restrictions

Let $W$ be a Brownian motion on some probability space $(\Omega, \mathcal{F}, P)$. Let $\mathbb{F}^W$ be the filtration generated by $W$ and let $X$ be a process that is progressively measurable ...
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17 views

conditional expectation of a sum of two independent Ornstein-Uhlenbeck type processes

Consider two independent processes of discrete Ornstein-Uhlenbeck type, $X_t$ and $Y_t$: \begin{eqnarray*} dX_t&=&\theta_x(\bar{X}-X_t) + \sigma_xdB_{xt}\\ dY_t&=&\theta_y(\bar{Y}-Y_t) ...
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1answer
27 views

Exponential Martingales - Properties

This question relates to the exponential martingale, \begin{align} Y(t) = \exp\left(-\int_{0}^{t} \lambda(s)\,dW(s) - \tfrac{1}{2} \int_{0}^{t} \lambda^2(s)\,ds \right) \end{align} and specifically ...
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1answer
32 views

How to understand the definition of weak convergence of stochastic processes

I have some problems with the definition of $\textit{weak convergence of stochastic processes}$. To ask my question, we start with two well-known definitions corresponding to measures and random ...
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25 views

Why do we need to declare a probability measure for the definition of stochastic processes?

Let $(\Omega,\mathcal{A})$ be a measurable space $E$ be a Polish space and $\mathcal{E}$ be the Borel-$\sigma$-algebra on $E$ $I\subseteq\mathbb{R}$ $X_t$ be measurable with respect to ...
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1answer
55 views

Martingale representation theorem application

Let $X = exp(W_{T/2}+W_T)$. I try to figure the adapted process $g(s)$ such that according to the MRT we have $$X = \mathbb{E}[X]+\int^T_0 g_s dW_s$$. I can figure out $X = exp(2W_{T/2}+W_{T-T/2})$ ...
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25 views

p.d.f. of a position variable from stochastic velocity p.d.f.

I have a stochastic process, $v(t)$, that represents a velocity, and has a known probability distribution function $f(x,t)$ which is time-varying. I am interested to acquire a probability ...
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27 views

Does this sequence converge? If yes, what is the limit?

Assume $\{k_n\}_{n\geq 0}$ a sequence of natural numbers such that $k_0=0$, $k_n\leq k_{n+1}\leq k_n+1$, and $\lim_{n\rightarrow\infty} \frac{k_n}{n}=\alpha\in(0,1)$. So $\{k_n\}$ is an ...
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1answer
25 views

proving independence of stochastic integrals

Does anyone know how to show that the stochastic integrals \begin{equation} \bigg\{ \int_0^1 \cos \Big[ (n- \frac{1}{2}) \pi t \Big] \,dW_t \bigg\}_{n \in \mathbb{N}} \end{equation} are ...
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1answer
17 views

Distribution of $\int^T_t \sigma (T-u)dW_u$ where $W_t$ is a Brownian motion

I am trying to find the distribution of $\int^T_t \sigma (T-u)dW_u$ where $W_t$ is a Brownian motion. One (very hand-wavey) way is to assume a priori that it is Normally distributed. Then one can ...
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1answer
13 views

Asymptotic stopping time for a ball-drawing problem

Take two different boxes, one with $N$ red balls and one with $N$ blue balls. Remove balls one at a time from either box with equal probability. When only one color is left, the (expected value of ...
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1answer
14 views

Construction of Probability Generating Function in Branching Process?

So I'm trying to construct a probability generating function for the following scenario: 1/5 of a rabbit population does not reproduce. 4/5 have 3 offspring each, and the probability of male or ...
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1answer
23 views

Properties of brownian motion

I was doing some revision and had an admittedly elementary question. My lecture notes say, the following are properties of Brownian Motion {$B_t$} (Normal or Gaussian increments) For all $s < t, ...
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1answer
30 views

Exponential Martingales

This is a two-part question concerning exponential martingales. It is stated that an application of Ito's lemma to \begin{align} \rho_t = \exp\left[-\int_{0}^{t} \lambda_s\,dW_s - ...
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1answer
32 views

Convergence of exponential Brownian martingale to zero almost surely

Define the exponential Brownian martingale as $N_t = \exp\left\{a W_t - \frac12 a^2 t \right\}$ which is a martingale with respect to the natural filtration of $W$ which stands for a standard Brownian ...
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27 views

Is there a name for this stochastic process?

Let $(\Omega,\mathscr{F},P)$ be a probability space and $\{X_n\}_{n\geq 1}$ be a stochastic process. Assume each $X_n$ only takes two values $0$ or $1$, i.e., $X_n:\Omega\rightarrow \{0,1\}$. Of ...
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12 views

Poisson Processes question

. Let {N(t) : t ≥ 0} be a Poisson process with rate λ > 0. Let Y be a random variable independent of (N(t)), such that Y = 1 with probability 1/2 and Y = −1 with probability 1/2. We define the new ...
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1answer
11 views

Poisson Processes - What is the distribution of the number of arrivals $Z$ happening in the random interval of time $[0,T]$?

Let $\{N(t) : t \geq 0\}$ be a Poisson process with rate $λ$, and $Z$ represent the number of arrivals in the interval of time $[0,t]$. Let $T$ be a random variable, exponentially distributed with ...
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1answer
32 views

Distribution of a transformed Brownian motion

Let $W$ be a standard Brownian motion. From an earlier proven result I know that $N_t = \exp\left\{a W_t - \frac12 a^2 t \right\}$ defines a martingale on the natural filtration of $W$ for all $a \in ...
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12 views

Book recommendation needed: asymptotic behavior of non-stationary Markov chain

Is there any stochastic process textbook which covers some standard results for non-stationary Markov chain? For my purpose, countable state space is enough. Thanks!
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1answer
28 views

Calculate $\mathbb{E}(T^2)$ and $\mathbb{E}(\int_0^T X_s \,d s)$ for exit time $T$ of Brownian motion $(X_t)_{t \geq 0}$

Let $T$ be the exit time of from the interval $[-b,a]$ of a standard Brownian Motion $X_t$, then how would we go about calculating the following two expectations: $E[T^2]$ (and) $E[\int_0^T X_tds]$? ...
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0answers
29 views

Measurability of the event that Brownian motion hits a given set

Let $W$ be a Brownian motion in $\mathbb{R}^{2}$ on a probability space $\left(\Omega,\mathcal{F},\mathbb{P}\right)$ . Let us assume $\mathcal{F}$ is the sigma-algebra on the path space ...
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7 views

Point processes that are not Cox?

Can some provide examples of point processes that are not Cox? A Cox process is a doubly stochastic poisson process with random intensity.
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1answer
76 views

Prove $\mathbb{P} \left( \sup_{t \geq 0} M_t > x \mid \mathcal{F}_0 \right) = 1 \wedge \frac{M_0}{x}$ for a martingale $(M_t)_{t \geq 0}$

Let $M$ be a positive, continuous martingale that converges a.s. to zero as $t$ tends to infinity. I now want to prove that for every $x>0$ $$ P\left( \sup_{t \geq 0 } M_t > x \mid \mathcal{F}_0 ...
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1answer
49 views

Supremal distribution of positive continuous martingale, which converges to zero a.s.

So the question is as follows: Let $M$ be a positive continous martingale, converging a.s. to zero as $t \rightarrow \infty$. Prove that for every $x>0$: $\mathbb{P}\{\sup_{\{t \geq 0 \}} M_t > ...
3
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1answer
38 views

Reference request for stochastic process

I studied the book, "Probability, Random Variables and Random Signal Principles" by Peyton Peebles. And I am a little bit familiar with statistical analysis like signal estimation and detection. In ...
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1answer
35 views

If two stochastic processes are modifications of each other and almost surely continuous from the right, then they are undistinguishable

Let $(\Omega,\mathcal{A},\operatorname{P})$ be a probability space $I\subseteq\mathbb{R}$ $E$ be a metric space and $\mathcal{E}:=\mathcal{B}(E)$ be the Borel-$\sigma$-algebra on $E$ $X:=(X_t)_{t\in ...
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0answers
16 views

Defining a stochastic process indexed by angle

I have a random closed curve of the form $(\theta,r_\theta)$, where $\theta\in [0,2\pi]$, is the counter clockwise angle from the x-axis and $r_\theta$ is the radial distance from the origin ...
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1answer
34 views

Adaptive gambler's ruin problem

Suppose in the gambler's ruin problem that the probability of winning a bet depends on the gambler's present fortune. Specifically, suppose that $p_{i}$ is the probability that the gambler wins a bet ...
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1answer
28 views

Men and Women enter a supermarket according to independent poisson process (stochastic process) [closed]

Men and Women enter a supermarket according to independent poisson processes having respective rates of two and four per minute. a) Starting at an arbitrary time, what is the probability that at ...
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Variance of interarrival time of events [closed]

As shown in the figure, in this problem, there are three types of events where events of each type occur independently. The inter-arrival time distribution between events of the same type is an ...
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1answer
56 views

Generating the Borel $\sigma$-algebra on $C([0,1])$

We put $S=C([0,1])$ (the collection of continuous real functions on $[0,1]$), equipped with the metric $d(f,g)=\sup_{x\in[0,1]}|f(x)-g(x)|$, and let $\mathcal{B}(S)$ be the Borel $\sigma$-algebra on ...
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2answers
47 views

How to prove $E[e^{e^y}]=\infty$? y is a normal random variable

The question is, given $Y\sim N(\mu,\sigma^2)$, how to prove$E[e^{e^Y}]=\infty$? I tried to look Y as some kind of Ito's process and apply Ito's formula to it but it doesn't make sense. Next I tried ...
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1answer
37 views

Distribution of $\| W_t \|^2_{L^2([0,T])}$

Motivation: consider the SDE $$dX_t = b(X_t) dt + \sqrt{\varepsilon} dW_t. \tag1$$ Consider the action, defined by $$S(\phi)=\int_0^T |\phi'(t)-b(\phi(t))|^2 dt$$ if $\phi \in H^1([0,T])$ and ...
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1answer
52 views

Show that a Markov Chain is ergodic

Let $Y_n$ be iid random variables with values 1,2,3..n so that $P[Y_i=j]=p_j>0$, where $i\leq1$ and $1\leq j\leq n$. I think I managed to show that $Y_n$ is a Markov chain using the definition, ...
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1answer
35 views

Stronger version of Markov Chain

I have just started looking into the concept of Markov chains and I was wondering if anyone could help me with this problem. Let $X_1, X_2, ...$ be a Markov chain with the state space $S$. I need ...
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1answer
31 views

Process adapted to Filtration [closed]

Here is the definition I have been given : A process $(X_t)$ is adapted to a filtration $(\mathcal F_t)$ if $X_t$ is $F_t$ measurable, for all t > 0 , i.e : $X_t^-1 (\mathcal B)$ belong to ...
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22 views

Convergence in distribution of BM started in (x,y) to BM started in (0,0)

Let $B$ be a Brownian motion in $\mathbb{R}^{2}$ . Let $\mathbb{P}_{(x,y)}$ denote a probability measure under which $B$ is started at $(x,y)$ . Is it true in general that, for measurable set ...
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22 views

Simple Symmetric Random Walk [closed]

Use Hint: Show first that for any random variable N with range {0,1,...},